arXiv subjects

Polly Y. Yu

Publications and source records attributed to Polly Y. Yu.

15 recordsLinked to original sources

arXiv2024

Generalized Lotka-Volterra Systems and Complex Balanced Polyexponential Systems

We study the global stability of generalized Lotka-Volterra systems with generalized polynomial right-hand side, without restrictions on the number of variables or the polynomial degree, including negative and non-integer degree. We introduce polyexponential dynamical systems, which are equivalent to the generalized Lotka-Volterra systems, and we use an analogy to the theory of mass-action kinetics to define and analyze complex balanced polyexponential systems, and implicitly analyze complex balanced generalized Lotka-Volterra systems. We prove that complex balanced generalized Lotka-Volterra systems have globally attracting states, up to standard conservation relations, which become linear for the associated polyexponential systems. In particular, complex balanced generalized Lotka-Volterra systems cannot give rise to periodic solutions, chaotic dynamics, or other complex dynamical behaviors. We describe a simple sufficient condition for complex balance in terms of an associated graph structure, and we use it to analyze specific examples.

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arXiv2024

Maximum likelihood estimation of log-affine models using detailed-balanced reaction networks

A fundamental question in the field of molecular computation is what computational tasks a biochemical system can carry out. In this work, we focus on the problem of finding the maximum likelihood estimate (MLE) for log-affine models. We revisit a construction due to Gopalkrishnan of a mass-action system with the MLE as its unique positive steady state, which is based on choosing a basis for the kernel of the design matrix of the model. We extend this construction to allow for any finite spanning set of the kernel, and explore how the choice of spanning set influences the dynamics of the resulting network, including the existence of boundary steady states, the deficiency of the network, and the rate of convergence. In particular, we prove that using a Markov basis as the spanning set guarantees global stability of the MLE steady state.

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arXiv2024

A necessary condition for non-monotonic dose response, with an application to a kinetic proofreading model -- Extended version

Steady state nonmonotonic ("biphasic") dose responses are often observed in experimental biology, which raises the control-theoretic question of identifying which possible mechanisms might underlie such behaviors. It is well known that the presence of an incoherent feedforward loop (IFFL) in a network may give rise to a nonmonotonic response. It has been conjectured that this condition is also necessary, i.e. that a nonmonotonic response implies the existence of an IFFL. In this paper, we show that this conjecture is false, and in the process prove a weaker version: that either an IFFL must exist or both a positive feedback loop and a negative feedback loop must exist. Towards this aim, we give necessary and sufficient conditions for when minors of a symbolic matrix have mixed signs. Finally, we study in full generality when a model of immune T-cell activation could exhibit a steady state nonmonotonic dose response.

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arXiv2022

Global stability of perturbed complex-balanced systems

A class of polynomial dynamical systems called complex-balanced are locally stable and conjectured to be globally stable. In general, complex-balancing is not a robust property, i.e., small changes in parameter values may result in the loss of the complex-balanced property. We show that robustly permanent complex-balanced systems are globally stable even after the rate constants have been perturbed.

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arXiv2022

An algorithm for finding weakly reversible deficiency zero realizations of polynomial dynamical systems

Systems of differential equations with polynomial right-hand sides are very common in applications. On the other hand, their mathematical analysis is very challenging in general, due to the possibility of complex dynamics: multiple basins of attraction, oscillations, and even chaotic dynamics. Even if we restrict our attention to mass-action systems, all of these complex dynamical behaviours are still possible. On the other hand, if a polynomial dynamical system has a weakly reversible deficiency zero ($WR_0$) realization, then its dynamics is known to be remarkably simple: oscillations and chaotic dynamics are ruled out and, up to linear conservation laws, there exists a single positive steady state, which is asymptotically stable. Here we describe an algorithm for finding $WR_0$ realizations of polynomial dynamical systems, whenever such realizations exist.

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arXiv2022

The disguised toric locus and affine equivalence of reaction networks

Under the assumption of mass-action kinetics, a dynamical system may be induced by several different reaction networks and/or parameters. It is therefore possible for a mass-action system to exhibit complex-balancing dynamics without being weakly reversible or satisfying toric constraints on the rate constants; such systems are called disguised toric dynamical systems. We show that the parameters that give rise to such systems are preserved under invertible affine transformations of the network. We also consider the dynamics of arbitrary mass-action systems under affine transformations, and show that there is a bijection between their sets of positive steady states, although their qualitative dynamics can differ substantially.

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arXiv2021

A graph-theoretic condition for delay stability of reaction systems

Delay mass-action systems provide a model of chemical kinetics when past states influence the current dynamics. In this work, we provide a graph-theoretic condition for delay stability, i.e., linear stability independent of both rate constants and delay parameters. In particular, the result applies when the system has no delay, implying asymptotic stability for the ODE system. The graph-theoretic condition is about cycles in the directed species-reaction graph of the network, which encodes how different species in the system interact.

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arXiv2021

Uniqueness of weakly reversible and deficiency zero realizations of dynamical systems

A reaction network together with a choice of rate constants uniquely gives rise to a system of differential equations, according to the law of mass-action kinetics. On the other hand, different networks can generate the same dynamical system under mass-action kinetics. Therefore, the problem of identifying "the" underlying network of a dynamical system is not well-posed, in general. Here we show that the problem of identifying an underlying weakly reversible deficiency zero network is well-posed, in the sense that the solution is unique whenever it exists. This can be very useful in applications because from the perspective of both dynamics and network structure, a weakly reversibly deficiency zero ($\textit{WR}_\textit{0}$) realization is the simplest possible one. Moreover, while mass-action systems can exhibit practically any dynamical behavior, including multistability, oscillations, and chaos, $WR_0$ systems are remarkably stable for any choice of rate constants: they have a unique positive steady state within each invariant polyhedron, and cannot give rise to oscillations or chaotic dynamics. We also prove that both of our hypotheses (i.e., weak reversibility and deficiency zero) are necessary for uniqueness.

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arXiv2021

Single-Target Networks

We characterize the dynamics of all single-target networks under mass-action kinetics: either the system is (i) globally stable for all choice of rate constants (in fact, dynamically equivalent to a detailed-balanced system) or (ii) has no positive steady states for any choice of rate constants and all trajectories must converge to the boundary of the positive orthant or to infinity. Moreover, global stability occurs if and only if the target vertex of the network is in the relative interior of the convex hull of the source vertices.

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arXiv2020

Autocatalytic systems and recombination: a reaction network perspective

Autocatalytic systems are very often incorporated in the "origin of life" models, a connection that has been analyzed in the context of the classical hypercycles introduced by Manfred Eigen. We investigate the dynamics of certain networks called bimolecular autocatalytic systems. In particular, we consider the dynamics corresponding to the relative populations in these networks, and show that they can be analyzed by studying well-chosen autonomous polynomial dynamical systems. Moreover, we find that one can use results from reaction network theory to prove persistence and permanence of several types of bimolecular autocatalytic systems called autocatalytic recombination networks.

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arXiv2020

Weakly reversible mass-action systems with infinitely many positive steady states

We show that weakly reversible mass-action systems can have a continuum of positive steady states, coming from the zeroes of a multivariate polynomial. Moreover, the same is true of systems whose underlying reaction network is reversible and has a single connected component. In our construction, we relate operations on the reaction network to the multivariate polynomial occurring as a common factor in the system of differential equations.

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arXiv2020

Delay stability of reaction systems

Delay differential equations are used as a model when the effect of past states has to be taken into account. In this work we consider delay models of chemical reaction networks with mass action kinetics. We obtain a sufficient condition for absolute delay stability of equilibrium concentrations, i.e., local asymptotic stability independent of the delay parameters. Several interesting examples on sequestration networks with delays are presented.

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arXiv2019

An efficient characterization of complex-balanced, detailed-balanced, and weakly reversible systems

Very often, models in biology, chemistry, physics, and engineering are systems of polynomial or power-law ordinary differential equations, arising from a reaction network. Such dynamical systems can be generated by many different reaction networks. On the other hand, networks with special properties (such as reversibility or weak reversibility) are known or conjectured to give rise to dynamical systems that have special properties: existence of positive steady states, persistence, permanence, and (for well-chosen parameters) complex balancing or detailed balancing. These last two are related to thermodynamic equilibrium, and therefore the positive steady states are unique and stable. We describe a computationally efficient characterization of polynomial or power-law dynamical systems that can be obtained as complex-balanced, detailed-balanced, weakly reversible, and reversible mass-action systems.

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arXiv2019

A generalization of Birch's theorem and vertex-balanced steady states for generalized mass-action systems

Mass-action kinetics and its generalizations appear in mathematical models of (bio-)chemical reaction networks, population dynamics, and epidemiology. The dynamical systems arising from directed graphs are generally non-linear and difficult to analyze. One approach to studying them is to find conditions on the network which either imply or preclude certain dynamical properties. For example, a vertex-balanced steady state for a generalized mass-action system is a state where the net flux through every vertex of the graph is zero. In particular, such steady states admit a monomial parametrization. The problem of existence and uniqueness of vertex-balanced steady states can be reformulated in two different ways, one of which is related to Birch's theorem in statistics, and the other one to the bijectivity of generalized polynomial maps, similar to maps appearing in geometric modelling. We present a generalization of Birch's theorem, by providing a sufficient condition for the existence and uniqueness of vertex-balanced steady states.

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arXiv2018

Mathematical Analysis of Chemical Reaction Systems

The use of mathematical methods for the analysis of chemical reaction systems has a very long history, and involves many types of models: deterministic versus stochastic, continuous versus discrete, and homogeneous versus spatially distributed. Here we focus on mathematical models based on deterministic mass-action kinetics. These models are systems of coupled nonlinear differential equations on the positive orthant. We explain how mathematical properties of the solutions of mass-action systems are strongly related to key properties of the networks of chemical reactions that generate them, such as specific versions of reversibility and feedback interactions.

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